为生成模型构建通用几何框架，实现高效高密度路径规划。
A Likely Geometry of Generative Models
- 提出基于数据分布约束的类测地线曲线，无需额外训练。
- 最短路径由常微分方程刻画，对应特定黎曼度量下的测地线。
- 算法可高效计算最短路径与广义弗雷歇均值，适用于多模型多数据集。
生成模型的几何结构是插值、模型分析的基础。然而，大多数生成模型缺乏普适的几何定义，需依赖对模型或数据维度的限制性假设。本文构建了一种兼容多种度量和概率分布的通用几何框架，用于分析无需额外训练的生成模型。我们考虑在合适数据分布上受限的类测地线曲线，旨在指向生成模型学习到的高密度区域。该方法被形式化为（伪）度量，并证明其等价于黎曼流形上的牛顿系统。我们证明，该框架下的最短路径可通过常微分方程表征，局部对应于特定黎曼度量下的测地线。数值上，我们提出一种新算法，可高效计算最短路径与广义弗雷歇均值。定量实验表明，相比基线方法，该度量下的曲线在多个模型和数据集上能更有效地穿越更高密度区域。
原文摘要 · Abstract (English)
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets.
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